A tangent or normal to a curve is a line, taking the formwhere
is the gradient and
is the intercept. Given a function
we can find the gradient at
by finding the gradient function
and substituting the value
into this expression. Sometimes however we don't have
so
is not given explicitly as a function of
In these cases typically we have to differentiate implicitly and find
as a function of both
and
and then substitute a point
into the expression for
to find the gradient at that point. Finally substitute into the equation
to find the equation of the line.
Example: Find the equation of the tangent to the curveat the point
We differentiate implicitly to getThe gradient at the point
is
Example: Find the equation of the tangent to the curveat the point
We differentiate implicitly to getWe have to make
the subject of this equation.
The gradient at the point is
Example: Find the equation of the normal to the curveat the point
We differentiate implicitly to getWe have to make
the subject of this equation.
The gradient at the point is